Summary: Notes on a formalization of the
prime number theorem
Jeremy Avigad
September 10, 2004
1 Introduction
On September 6, 2004, using the Isabelle proof assistant, I verified the following
statement:
(%x. pi x * ln (real x) / (real x)) ----> 1
The system thereby confirmed that the prime number theorem is a consequence
of the axioms of higher-order logic together with an axiom asserting the exis-
tence of an infinite set. All told, our number theory session, including the proof
of the prime number theorem and supporting libraries, constitutes 673 pages of
proof scripts, or roughly 30,000 lines. This count includes about 65 pages of
elementary number theory that we had at the outset, developed by Larry Paul-
son and others; also about 50 pages devoted to a proof of the law of quadratic
reciprocity and properties of Euler's function, neither of which are used in
the proof of the prime number theorem. The page count does not include the
basic HOL library, or properties of the real numbers that we obtained from the
HOL-Complex library.
The formalization was a collaborative effort. David Gray developed a sub-